Body Fat Percentage Presentation

Group 013E01

Joanne Lim, John Fu, Haoyuan Gao, Zhenchen Yi

Data Description

  • Accurately body fat measurement is vital but often inconvenient and expensive
  • Can we use simpler measurements like height, weight, and circumferences for estimates?
  • The data set bodyfat contains body fat percentages and other related measurements for 250 men. These measurements, which include height, weight and various body circumferences, were collected to explore alternatives to underwater body fat assessments.
Rows: 250
Columns: 16
$ Density <dbl> 1.0708, 1.0853, 1.0414, 1.0751, 1.0340, 1.0502, 1.0549, 1.0704…
$ Pct.BF  <dbl> 12.3, 6.1, 25.3, 10.4, 28.7, 20.9, 19.2, 12.4, 4.1, 11.7, 7.1,…
$ Age     <int> 23, 22, 22, 26, 24, 24, 26, 25, 25, 23, 26, 27, 32, 30, 35, 35…
$ Weight  <dbl> 154.25, 173.25, 154.00, 184.75, 184.25, 210.25, 181.00, 176.00…
$ Height  <dbl> 67.75, 72.25, 66.25, 72.25, 71.25, 74.75, 69.75, 72.50, 74.00,…
$ Neck    <dbl> 36.2, 38.5, 34.0, 37.4, 34.4, 39.0, 36.4, 37.8, 38.1, 42.1, 38…
$ Chest   <dbl> 93.1, 93.6, 95.8, 101.8, 97.3, 104.5, 105.1, 99.6, 100.9, 99.6…
$ Abdomen <dbl> 85.2, 83.0, 87.9, 86.4, 100.0, 94.4, 90.7, 88.5, 82.5, 88.6, 8…
$ Waist   <dbl> 33.54331, 32.67717, 34.60630, 34.01575, 39.37008, 37.16535, 35…
$ Hip     <dbl> 94.5, 98.7, 99.2, 101.2, 101.9, 107.8, 100.3, 97.1, 99.9, 104.…
$ Thigh   <dbl> 59.0, 58.7, 59.6, 60.1, 63.2, 66.0, 58.4, 60.0, 62.9, 63.1, 59…
$ Knee    <dbl> 37.3, 37.3, 38.9, 37.3, 42.2, 42.0, 38.3, 39.4, 38.3, 41.7, 39…
$ Ankle   <dbl> 21.9, 23.4, 24.0, 22.8, 24.0, 25.6, 22.9, 23.2, 23.8, 25.0, 25…
$ Bicep   <dbl> 32.0, 30.5, 28.8, 32.4, 32.2, 35.7, 31.9, 30.5, 35.9, 35.6, 32…
$ Forearm <dbl> 27.4, 28.9, 25.2, 29.4, 27.7, 30.6, 27.8, 29.0, 31.1, 30.0, 29…
$ Wrist   <dbl> 17.1, 18.2, 16.6, 18.2, 17.7, 18.8, 17.7, 18.8, 18.2, 19.2, 18…

Check Linearity

Null and Full Model


Call:
lm(formula = Pct.BF ~ ., data = bodyfat)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.3746 -0.3725 -0.1157  0.2358 15.0629 

Coefficients: (1 not defined because of singularities)
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  4.494e+02  1.154e+01  38.961   <2e-16 ***
Density     -4.098e+02  8.384e+00 -48.876   <2e-16 ***
Age          1.395e-02  9.721e-03   1.435    0.153    
Weight       1.527e-02  2.015e-02   0.758    0.449    
Height      -1.558e-02  5.752e-02  -0.271    0.787    
Neck        -1.653e-02  7.084e-02  -0.233    0.816    
Chest        1.790e-02  3.259e-02   0.549    0.583    
Abdomen      1.833e-02  3.286e-02   0.558    0.578    
Waist               NA         NA      NA       NA    
Hip          2.537e-02  4.391e-02   0.578    0.564    
Thigh       -2.107e-02  4.421e-02  -0.476    0.634    
Knee        -1.657e-02  7.366e-02  -0.225    0.822    
Ankle       -8.160e-02  6.616e-02  -1.233    0.219    
Bicep       -5.256e-02  5.132e-02  -1.024    0.307    
Forearm      1.405e-02  6.229e-02   0.225    0.822    
Wrist       -1.883e-02  1.640e-01  -0.115    0.909    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.276 on 235 degrees of freedom
Multiple R-squared:  0.9777,    Adjusted R-squared:  0.9763 
F-statistic: 734.4 on 14 and 235 DF,  p-value: < 2.2e-16

Backward and Forward stepwise selection


Call:
lm(formula = Pct.BF ~ Density + Age + Abdomen, data = bodyfat)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.2913 -0.3576 -0.0911  0.2319 15.4601 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  4.424e+02  8.738e+00  50.626  < 2e-16 ***
Density     -4.065e+02  7.279e+00 -55.844  < 2e-16 ***
Age          1.182e-02  6.579e-03   1.796   0.0737 .  
Abdomen      5.761e-02  1.332e-02   4.326 2.21e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.26 on 246 degrees of freedom
Multiple R-squared:  0.9772,    Adjusted R-squared:  0.9769 
F-statistic:  3513 on 3 and 246 DF,  p-value: < 2.2e-16

Call:
lm(formula = Pct.BF ~ Density + Abdomen + Age, data = bodyfat)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.2913 -0.3576 -0.0911  0.2319 15.4601 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  4.424e+02  8.738e+00  50.626  < 2e-16 ***
Density     -4.065e+02  7.279e+00 -55.844  < 2e-16 ***
Abdomen      5.761e-02  1.332e-02   4.326 2.21e-05 ***
Age          1.182e-02  6.579e-03   1.796   0.0737 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.26 on 246 degrees of freedom
Multiple R-squared:  0.9772,    Adjusted R-squared:  0.9769 
F-statistic:  3513 on 3 and 246 DF,  p-value: < 2.2e-16

log-linear


Call:
lm(formula = log(Pct.BF) ~ Density + Age + Weight + Height + 
    Neck + Chest + Abdomen + Waist + Hip + Thigh + Knee + Ankle + 
    Bicep + Forearm + Wrist, data = bodyfat)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.83722 -0.06872  0.03803  0.10425  1.10099 

Coefficients: (1 not defined because of singularities)
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  30.327682   1.893223  16.019  < 2e-16 ***
Density     -28.194739   1.390322 -20.279  < 2e-16 ***
Age           0.003161   0.001595   1.982  0.04865 *  
Weight       -0.002840   0.003325  -0.854  0.39396    
Height        0.028504   0.009489   3.004  0.00296 ** 
Neck         -0.004012   0.011622  -0.345  0.73027    
Chest         0.002934   0.005412   0.542  0.58830    
Abdomen      -0.002992   0.005408  -0.553  0.58062    
Waist               NA         NA      NA       NA    
Hip          -0.007137   0.007208  -0.990  0.32311    
Thigh         0.018438   0.007295   2.528  0.01215 *  
Knee          0.001581   0.012085   0.131  0.89603    
Ankle        -0.011140   0.010860  -1.026  0.30605    
Bicep         0.006681   0.008440   0.792  0.42940    
Forearm      -0.001098   0.010233  -0.107  0.91463    
Wrist         0.021332   0.026949   0.792  0.42941    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2094 on 234 degrees of freedom
Multiple R-squared:  0.863, Adjusted R-squared:  0.8548 
F-statistic: 105.3 on 14 and 234 DF,  p-value: < 2.2e-16

linear-log

We observed that the relationship between Neck, Abdomen, Bicep, Forearm, Ankle and Pct.BF has a non-linear relationship, so we will transform these variable.


Call:
lm(formula = Pct.BF ~ Density + Age + Weight + Height + log(Neck) + 
    Chest + log(Abdomen) + Waist + Hip + Thigh + Knee + log(Ankle) + 
    log(Bicep) + log(Forearm) + Wrist, data = bodyfat)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.3203 -0.3693 -0.0711  0.2796 15.0887 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)   4.468e+02  4.196e+01  10.648   <2e-16 ***
Density      -4.136e+02  8.361e+00 -49.466   <2e-16 ***
Age           1.374e-02  9.564e-03   1.437    0.152    
Weight        9.328e-03  2.033e-02   0.459    0.647    
Height       -8.618e-04  5.967e-02  -0.014    0.988    
log(Neck)    -8.816e-01  2.618e+00  -0.337    0.737    
Chest         3.469e-02  3.241e-02   1.070    0.286    
log(Abdomen)  3.997e+00  1.194e+01   0.335    0.738    
Waist        -8.484e-02  3.442e-01  -0.247    0.805    
Hip           3.187e-02  4.328e-02   0.736    0.462    
Thigh        -8.900e-03  4.452e-02  -0.200    0.842    
Knee         -1.567e-02  7.319e-02  -0.214    0.831    
log(Ankle)   -2.118e+00  1.673e+00  -1.266    0.207    
log(Bicep)   -1.957e+00  1.630e+00  -1.200    0.231    
log(Forearm)  7.233e-01  1.694e+00   0.427    0.670    
Wrist         2.292e-02  1.621e-01   0.141    0.888    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.252 on 233 degrees of freedom
Multiple R-squared:  0.9782,    Adjusted R-squared:  0.9768 
F-statistic: 697.1 on 15 and 233 DF,  p-value: < 2.2e-16

log-log


Call:
lm(formula = log(Pct.BF) ~ Density + Age + Weight + Height + 
    log(Neck) + Chest + log(Abdomen) + Waist + Hip + Thigh + 
    Knee + log(Ankle) + log(Bicep) + log(Forearm) + Wrist, data = bodyfat)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.72641 -0.06183  0.01085  0.07258  1.02223 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)   -9.571417   6.401341  -1.495   0.1362    
Density      -27.395617   1.275625 -21.476  < 2e-16 ***
Age            0.002563   0.001459   1.757   0.0803 .  
Weight         0.002424   0.003102   0.782   0.4352    
Height         0.008675   0.009104   0.953   0.3416    
log(Neck)     -0.599698   0.399458  -1.501   0.1346    
Chest          0.002684   0.004944   0.543   0.5878    
log(Abdomen)  12.409951   1.821969   6.811 8.16e-11 ***
Waist         -0.354239   0.052507  -6.746 1.19e-10 ***
Hip           -0.002133   0.006602  -0.323   0.7470    
Thigh          0.008926   0.006793   1.314   0.1901    
Knee          -0.007167   0.011166  -0.642   0.5216    
log(Ankle)    -0.420123   0.255291  -1.646   0.1012    
log(Bicep)     0.071632   0.248740   0.288   0.7736    
log(Forearm)  -0.135995   0.258495  -0.526   0.5993    
Wrist          0.019089   0.024739   0.772   0.4411    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.1911 on 233 degrees of freedom
Multiple R-squared:  0.8864,    Adjusted R-squared:  0.8791 
F-statistic: 121.2 on 15 and 233 DF,  p-value: < 2.2e-16

nothing

  Full Backward Forward Log-Linear Linear-Log Log-Log
Predictors Estimates p Estimates p Estimates p Estimates p Estimates p Estimates p
(Intercept) 449.43 <0.001 442.38 <0.001 442.38 <0.001 30.33 <0.001 446.75 <0.001 -9.57 0.136
Density -409.76 <0.001 -406.49 <0.001 -406.49 <0.001 -28.19 <0.001 -413.59 <0.001 -27.40 <0.001
Age 0.01 0.153 0.01 0.074 0.01 0.074 0.00 0.049 0.01 0.152 0.00 0.080
Weight 0.02 0.449 -0.00 0.394 0.01 0.647 0.00 0.435
Height -0.02 0.787 0.03 0.003 -0.00 0.988 0.01 0.342
Neck -0.02 0.816 -0.00 0.730
Chest 0.02 0.583 0.00 0.588 0.03 0.286 0.00 0.588
Abdomen 0.02 0.578 0.06 <0.001 0.06 <0.001 -0.00 0.581
Hip 0.03 0.564 -0.01 0.323 0.03 0.462 -0.00 0.747
Thigh -0.02 0.634 0.02 0.012 -0.01 0.842 0.01 0.190
Knee -0.02 0.822 0.00 0.896 -0.02 0.831 -0.01 0.522
Ankle -0.08 0.219 -0.01 0.306
Bicep -0.05 0.307 0.01 0.429
Forearm 0.01 0.822 -0.00 0.915
Wrist -0.02 0.909 0.02 0.429 0.02 0.888 0.02 0.441
Neck [log] -0.88 0.737 -0.60 0.135
Abdomen [log] 4.00 0.738 12.41 <0.001
Waist -0.08 0.805 -0.35 <0.001
Ankle [log] -2.12 0.207 -0.42 0.101
Bicep [log] -1.96 0.231 0.07 0.774
Forearm [log] 0.72 0.670 -0.14 0.599
Observations 250 250 250 249 249 249
R2 / R2 adjusted 0.978 / 0.976 0.977 / 0.977 0.977 / 0.977 0.863 / 0.855 0.978 / 0.977 0.886 / 0.879
AIC 847.962 831.066 831.066 1353.555 836.172 1308.930